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Theorem oppgmnd 15150
Description: The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
Hypothesis
Ref Expression
oppgbas.1  |-  O  =  (oppg
`  R )
Assertion
Ref Expression
oppgmnd  |-  ( R  e.  Mnd  ->  O  e.  Mnd )

Proof of Theorem oppgmnd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppgbas.1 . . . 4  |-  O  =  (oppg
`  R )
2 eqid 2436 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2oppgbas 15147 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 11 . 2  |-  ( R  e.  Mnd  ->  ( Base `  R )  =  ( Base `  O
) )
5 eqidd 2437 . 2  |-  ( R  e.  Mnd  ->  ( +g  `  O )  =  ( +g  `  O
) )
6 eqid 2436 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
7 eqid 2436 . . . 4  |-  ( +g  `  O )  =  ( +g  `  O )
86, 1, 7oppgplus 15145 . . 3  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  R ) x )
92, 6mndcl 14695 . . . 4  |-  ( ( R  e.  Mnd  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
1093com23 1159 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
118, 10syl5eqel 2520 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  O
) y )  e.  ( Base `  R
) )
12 simpl 444 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  R  e.  Mnd )
13 simpr3 965 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
z  e.  ( Base `  R ) )
14 simpr2 964 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
y  e.  ( Base `  R ) )
15 simpr1 963 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  x  e.  ( Base `  R ) )
162, 6mndass 14696 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( z  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R ) ) )  ->  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) ) )
1712, 13, 14, 15, 16syl13anc 1186 . . . 4  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( z ( +g  `  R ) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R ) ( y ( +g  `  R ) x ) ) )
1817eqcomd 2441 . . 3  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( z ( +g  `  R ) ( y ( +g  `  R
) x ) )  =  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x ) )
198oveq1i 6091 . . . 4  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( ( y ( +g  `  R ) x ) ( +g  `  O
) z )
206, 1, 7oppgplus 15145 . . . 4  |-  ( ( y ( +g  `  R
) x ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
2119, 20eqtri 2456 . . 3  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
226, 1, 7oppgplus 15145 . . . . 5  |-  ( y ( +g  `  O
) z )  =  ( z ( +g  `  R ) y )
2322oveq2i 6092 . . . 4  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( x ( +g  `  O ) ( z ( +g  `  R
) y ) )
246, 1, 7oppgplus 15145 . . . 4  |-  ( x ( +g  `  O
) ( z ( +g  `  R ) y ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2523, 24eqtri 2456 . . 3  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2618, 21, 253eqtr4g 2493 . 2  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x ( +g  `  O ) y ) ( +g  `  O ) z )  =  ( x ( +g  `  O ) ( y ( +g  `  O ) z ) ) )
27 eqid 2436 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
282, 27mndidcl 14714 . 2  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  ( Base `  R
) )
296, 1, 7oppgplus 15145 . . 3  |-  ( ( 0g `  R ) ( +g  `  O
) x )  =  ( x ( +g  `  R ) ( 0g
`  R ) )
302, 6, 27mndrid 14717 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  R ) ( 0g
`  R ) )  =  x )
3129, 30syl5eq 2480 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  O ) x )  =  x )
326, 1, 7oppgplus 15145 . . 3  |-  ( x ( +g  `  O
) ( 0g `  R ) )  =  ( ( 0g `  R ) ( +g  `  R ) x )
332, 6, 27mndlid 14716 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  R ) x )  =  x )
3432, 33syl5eq 2480 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  O ) ( 0g
`  R ) )  =  x )
354, 5, 11, 26, 28, 31, 34ismndd 14719 1  |-  ( R  e.  Mnd  ->  O  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Mndcmnd 14684  oppgcoppg 15141
This theorem is referenced by:  oppgmndb  15151  oppggrp  15153  gsumwrev  15162  gsumzoppg  15539  gsumzinv  15540  oppgtmd  18127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-0g 13727  df-mnd 14690  df-oppg 15142
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